Here's my guess:
The speaker is Gollum. ("Hiker on his out and return trip" would be Bilbo "there and back again"). The location is Mordor. Gollum's interlocutor is Sauron. The question, not asked inversely, is "where is the Ring?" Which, from Sauron's point of view, is much the same as "where am I?" This scene is referenced, but never depicted, in the books.
UPDATE: I finally figured out, duh, what is meant by "a place without hands". Here's a revised guess:
The speaker is the One Ring, and the place is Bilbo's pocket.
RE-UPDATE: I had assumed all the riddles were classics and well-known, besides being in the book. My answer may be cheapened a little, because all these were known to me in advance. Anyway here are the riddle explanations:
The darkness which cannot be felt
Answer one of Gollum's riddles to Bilbo, which starts "it cannot be seen, cannot be felt". The answer is "dark".
The treasure in the lidless lockbox
Answer is "eggs", or as Gollum puts it, "eggses". An alternate form of this riddle reads in part "no doors are there to this stronghold / yet thieves break in and steal the gold".
dancing white horses with gums for socks:
Teeth. One of Bilbo's riddles for Gollum. "Thirty white horses upon a red hill..." I've had this riddle in memory since childhood.
The roots of the mountain which outgrows the trees
what has roots, goes up and up but never grows? A mountain. One of Gollum's riddles to Bilbo.
And one more:
"No legs lay on one leg, two legs sat by on three legs, four legs got some." I don't like to spoil this excellent riddle.
And then the final question, or the final 'riddle' Bilbo asks:
"What have I got in my pocket?" One of Gollum's incorrect guesses was "hands", giving the question's title.